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Computing the  -Selmer group of an elliptic curve
Author(s):
Z.
Djabri;
Edward
F.
Schaefer;
N.
P.
Smart
Journal:
Trans. Amer. Math. Soc.
352
(2000),
5583-5597.
MSC (2000):
Primary 11G05, 11Y99;
Secondary 14H52, 14Q05
Posted:
August 21, 2000
MathSciNet review:
1694286
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Abstract:
In this paper we explain how to bound the  -Selmer group of an elliptic curve over a number field  . Our method is an algorithm which is relatively simple to implement, although it requires data such as units and class groups from number fields of degree at most  . Our method is practical for  , but for larger values of  it becomes impractical with current computing power. In the examples we have calculated, our method produces exactly the  -Selmer group of the curve, and so one can use the method to find the Mordell-Weil rank of the curve when the usual method of  -descent fails.
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Additional Information:
Z.
Djabri
Affiliation:
Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, Kent, CT2 7NF, United Kingdom
Address at time of publication:
Riskcare, Piercy House, 7 Copthall Avenue, London EC2R 7NJ, United Kingdom
Email:
zmd1@ukc.ac.uk, zdjabri@riskcare.com
Edward
F.
Schaefer
Affiliation:
Department of Mathematics, Santa Clara University, Santa Clara, California 95053
Email:
eschaefe@math.scu.edu
N.
P.
Smart
Affiliation:
Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol, BS12 6QZ, United Kingdom
Address at time of publication:
Computer Science Department, Woodland Road, University of Bristol, Bristol, BS8 1UB, United Kingdom
Email:
nsma@hplb.hpl.hp.com, nigel@cs.bris.ac.uk
DOI:
10.1090/S0002-9947-00-02535-6
PII:
S 0002-9947(00)02535-6
Keywords:
Elliptic curves,
Mordell-Weil rank,
Selmer group
Received by editor(s):
October 28, 1998
Received by editor(s) in revised form:
February 26, 1999 and March 17, 1999
Posted:
August 21, 2000
Copyright of article:
Copyright
2000,
American Mathematical Society
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